Scattering, Clouds & Climate: A Short Workshop of Exploration

Mathematical Institute, University of Oxford

24th - 25th March 2014

Workshop aims

• Bring together mathematicians and atmospheric physicists researching scattering of electromagnetic radiation by atmospheric ice crystals.

• Identify areas where advances in mathematical methods may be useful in climate modelling applications.

• Nucleate future collaborative efforts.

Workshop format

• “Workshop” not a “conference”

• Talks 30 mins + 10-15 mins for questions/discussion

• Please ask/take questions during talks!

• Two scheduled discussion sessions

Programme for Monday

1300-1400 Lunch and registration

1400-1415 Welcome and introduction

1415-1545 Talks

1545-1615 Tea/coffee

1615-1745 Talks

1745-1830 Discussion

1830-1900 Free time

(opportunity to check in at Somerville College)

1900 Reception and dinner at Somerville College

Ice particles in clouds • Atmospheric ice crystals can form over a wide range of

temperatures – from just below 0°C to ≈-80°C

• Generally, the colder the cloud, the smaller the crystals (less water vapour available, inefficient dissipation of latent heat) – In coldest clouds crystals may be 100μm or smaller

– In warmer clouds crystals may grow to a few mm in size

• Simplest forms are solid ice hexagonal prisms…

… but not all ice crystals are that simple! From Atmospheric Halos by Walter Tape ISBN 0-87590-834-9

From www.snowcrystals.com

• Once large enough, the crystals start

to fall through the atmosphere. They

can collide with other crystals to

form large complex aggregates up to

1cm in size

• They can also collide with

supercooled droplets which

freeze on their surface

(eventually leading to hail)

1 mm

From Brandes et al 2008 J. Appl. Met. & Clim. http://dx.doi.org/10.1175/2008JAMC1869.1

From www.snowcrystals.com

Riming of an ice crystal in a supercooled wind tunnel, leading to graupel / small hail

Why care about E-M scattering by ice particles?

1. Radiative Transfer

– Light from the sun (“shortwave” radiation 0.2 – 4 μm wavelength) is scattered by ice clouds, some reflected back to space (cooling effect)

– Infrared light is emitted from the earth (“longwave” radiation 4 – 100 μm wavelength) – scattered and absorbed by ice clouds, some is scattered/reemitted back to the surface (warming effect)

– Overall effect on climate: longwave warming wins in a global average sense – but individual clouds may warm or cool us. Can we predict this from the number, size and shape of the crystals in the cloud?

Anthony Baran

Steven Dobbie

Joseph Ulanowski

Evelyn Hesse

Helen Smith

Paul Connolly

Juliet Pickering

Cathryn Fox

Robin Hogan

2. Remote Sensing

– Measure electromagnetic waves scattered by ice particles to infer the microphysics of the particles (number, size, shape)

– Two types:

• “Active” = radar, lidar. Generate a short pulse of light, and then listen for echo. Backscattered wave (power and polarisation) tells you something about the ice particles.

• “Passive” = radiometers. No transmission of waves, rely on either scattering of sunlight, or emission from particles

– Wavelengths from 10 cm to 0.3 μm

David Bebbington

Robin Hogan

Caroline Poulsen

Joseph Ulanowski

Chris Westbrook

Anthony Baran

Thorwald Stein

Why care about E-M scattering by ice particles?

Chilbolton radar - from www.stfc.ac.uk

MODIS satellite (NASA)

3. Optical effects

– Ice particles can scatter sunlight to form amazing optical phenomena (arcs and halos)

– Some effects well understood (eg circumzenithal arc)

– Some are not understood at all (eg “odd-radius” halos) – need to understand from scattering theory what kind of crystals could cause these effects

Joseph Ulanowski

Chris Westbrook

Steven Dobbie

Why care about E-M scattering by ice particles?

NASA Earth Science Picture of the Day, April 06, 2012

From www.atoptics.co.uk/halo/pyredens.htm

Requirements for scattering calculations • Particle span 𝐷 relative to wavelength 𝜆 spans range

from 𝐷 𝜆 ~0.001 − 1000 depending on the problem - no single technique can be applied over this range

• 3D Maxwell not 2D Helmholtz

• Need to be able to handle transmission (ice is a dielectric, not a conductor), and ideally absorption

• Want the far-field, single scattering pattern (cloud is dilute; multiple scattering can be important, but can be dealt with using this information and a RT model)

• At minimum need to handle a hexagonal prism. Ideally flexible geometry to deal with complex particles

• Mostly off-line calculations – then we integrate over size/shape distributions / create look up tables. So need to be able to solve a problem using a realistic computer spec, but doesn’t necessarily need to be super-fast

Typical tools • 𝐷 𝜆 ~0.001 − 10.

– Mie theory or T-matrix ~ rigorous derivation from Maxwell’s equations, computationally efficient, but practically very difficult to apply beyond highly symmetrical shapes such as spheroids (so approximate real shape by a simplification)

– Discrete Dipole Approximation (DDA) ~ treat scatterer as a matrix of small, polarisable dipoles. Physically intuitive, completely flexible geometry. Computationally expensive for large, intricate particles (CPU and memory).

– Less common approaches:

• Rayleigh-Gans approximation (aka Born approximation) ~ essentially first order version of DDA with no coupling between dipoles – reduces to simple sum of interference terms (Robin Hogan later)

• Fredholm Integral method (see David Bebbington’s talk later)

• Generalized Multiparticle Mie-Solution (group at Penn State)

Typical tools • 𝐷 𝜆 ~10 − 1000.

– Geometric optics ~ high frequency limit: trace light rays

along straight line paths using Monte Carlo approach.

Flexible geometry. Assumes all facets flat on scales of

order the wavelength (often not OK!). Not clear exactly

how to account for diffraction. Various kluges to deal

with these two problems. Simple and cheap on memory.

– Finite Difference Time Domain ~ very computationally

expensive, but more rigorous in the sense that are now

solving Maxwell’s equations proper. Need a very big

computational domain to simulate far-field, but fine mesh

for short wavelength. Only really practical at the lower

end of this 𝐷 𝜆 range.

Some possible collaboration points? • Development of new ideas / numerical tools to compute

scattering properties

– faster / less memory-intensive techniques (all 𝐷 𝜆 values)

– particular lack of options / rigour for 𝐷 𝜆 > 10

– ideas for how to deal with wavelength-scale “roughness” on particle surfaces at large 𝐷 𝜆

• Development / application of better software packages to do these calculations than are used in ice community at present

– eg BEM++ ?

• Error analysis / convergence of numerical tools – often we simply use a tool and assume the results to be accurate. Is this OK?

– Mathematical results for error bounds?

• Development of new measurement datasets to test the theoretical / numerical predictions (MICC, SID3, etc…)

Mathematical analysis of wave scattering

• Very mature field, but still an active research area in applied mathematics

• Acoustic, electromagnetic and elastic waves

• Time and frequency domains

• Model problem (frequency domain):

∆ + 𝑘2 u = 0

• What are people working on?

Analytical methods

• High frequency asymptotics (Geometrical Optics, Geometrical Theory of Diffraction)

• Canonical problems (wedges, cones, ray tangency, concave-convex transition)

• Inhomogeneous media:

– Composite materials

– Homogenisation

– Scattering at ice-water interface

David Abrahams Jon Chapman Samuel Groth John Hannay

David Hewett John Ockendon

Anthony Rawlins Valery Smyshlyaev

John Wettlaufer

Numerical methods

• Development of new FEM/BEM formulations • Error/stability analysis • Fast implementations (e.g. BEM++)

– Matrix compression (FMM, H-matrix) – Preconditioning – Domain decomposition – Parallelisation

• Rough surface scattering • Inhomogeneous media

• Traditionally a low/mid frequency approach, but high

frequency “hybrid numerical-asymptotic” methods are emerging

Timo Betcke Simon Chandler-Wilde

Andrew Gibbs Ivan Graham

Samuel Groth David Hewett

Stephen Langdon Valery Smyshlyaev

Some objectives

• Understand the key issues relating to ice scattering that currently limiting the accurate modelling of earth’s radiation balance

• Identify opportunities to apply emerging mathematical tools (computational and analytical)

• Identify “model” and “challenge” problems of relevance to application and of interest to mathematicians